Absolute Value

 “The Proving Tool — Empirical Geometric Proofs via Rotational and Radial Constructions”

Author: Deanna Rachel Sellers

Summary:

The Proving Tool is a visual-constructive system that uses layered radial and polygonal constructions to produce empirical geometric demonstrations of algebraic and trigonometric relationships. It encodes numeric relationships (for example, square roots and relationships involving π) as explicit, reproducible planar constructions: nested polygons, rotated squares, repeated triangular modules, and chord/arc partitions. The tool’s primary methodology is to translate algebraic relations into geometric scale/rotation/area identities that are measurable from the diagram itself, making the diagram both the demonstration and the datum.

Core components and mechanisms:

1. Radial Partition Engine — divides a circle into equal angular increments (e.g., 20, 18° increments) and places repeated triangular modules at each spoke to encode angle-based identities.

2. Rotation-Scale Module — constructs rotated congruent polygons (e.g., rotate a unit square 45° to produce a square whose side equals the diagonal √2) to visualize scale factors corresponding to roots.

3. Area-Based Evidence — for Pythagorean-type arguments, constructs squares on each triangle side so that measurable areas provide direct numeric verification (area of square on hypotenuse equals sum of area of leg-squares).

4. Recursive Nesting / Spiral Encoding — repeated nesting of polygons/spirals encodes sequences (e.g., Fibonacci-like growth or multiplicative relationships) and provides a visual mapping of series from unity outward.

5. Annotation & Measurement Layer — numerical labels, angle markers, and coordinate output are available so the diagram can be measured mathematically (coordinates, distances, areas) and not only inspected visually.

Why this is a novel/protectable approach:

While geometric proofs of classic constants (π, √2) exist historically, Proving Tool unifies several features in a reproducible engineering artifact: automated radial partitioning combined with an annotated measurement layer and an empirical-export format that allows any third party to numerically validate the visual proof. This integration—visual construct + reproducible numeric export + procedural animation—constitutes the inventive combination that differentiates it from a static diagram.